Algorithms and Data Structures Cheatsheet

We summarize the performance characteristics of classic algorithms and data structures for sorting, priority queues, symbol tables, and graph processing.

We also summarize some of the mathematics useful in the analysis of algorithms, including commonly encountered functions; useful formulas and approximations; properties of logarithms; asymptotic notations; and solutions to divide-and-conquer recurrences.

Sorting.

The table below summarizes the number of compares for a variety of sorting algorithms, as implemented in this textbook. It includes leading constants but ignores lower-order terms.

ALGORITHM CODE IN PLACE STABLE BEST AVERAGE WORST REMARKS

selection sort Selection.java ✔ ½ n² ½ n² ½ n² n exchanges;
quadratic in best case

insertion sort Insertion.java ✔ ✔ n ¼ n² ½ n² use for small or
partially-sorted arrays

bubble sort Bubble.java ✔ ✔ n ½ n² ½ n² rarely useful;
dominated by insertion sort

shellsort Shell.java ✔ n log₃ n unknown c n^3/2 tight code;
subquadratic

mergesort Merge.java ✔ ½ n log₂ n n log₂ n n log₂ n n log n guarantee;
stable

quicksort Quick.java ✔ n log₂ n 2 n ln n ½ n² n log n probabilistic guarantee;
fastest in practice

heapsort Heap.java ✔ n ^† 2 n log₂ n 2 n log₂ n n log n guarantee;
in place

^† n log₂ n if all keys are distinct

ALGORITHM	CODE	IN PLACE	STABLE	BEST	AVERAGE	WORST	REMARKS
selection sort	Selection.java	✔		½ n²	½ n²	½ n²	n exchanges; quadratic in best case
insertion sort	Insertion.java	✔	✔	n	¼ n²	½ n²	use for small or partially-sorted arrays
bubble sort	Bubble.java	✔	✔	n	½ n²	½ n²	rarely useful; dominated by insertion sort
shellsort	Shell.java	✔		n log₃ n	unknown	c n^3/2	tight code; subquadratic
mergesort	Merge.java		✔	½ n log₂ n	n log₂ n	n log₂ n	n log n guarantee; stable
quicksort	Quick.java	✔		n log₂ n	2 n ln n	½ n²	n log n probabilistic guarantee; fastest in practice
heapsort	Heap.java	✔		n ^†	2 n log₂ n	2 n log₂ n	n log n guarantee; in place
							^† n log₂ n if all keys are distinct

Priority queues.

The table below summarizes the order of growth of the running time of operations for a variety of priority queues, as implemented in this textbook. It ignores leading constants and lower-order terms. Except as noted, all running times are worst-case running times.

DATA STRUCTURE CODE INSERT DEL-MIN MIN DEC-KEY DELETE MERGE

array BruteIndexMinPQ.java 1 n n 1 1 n

binary heap IndexMinPQ.java log n log n 1 log n log n n

d-way heap IndexMultiwayMinPQ.java log_d n d log_d n 1 log_d n d log_d n n

binomial heap IndexBinomialMinPQ.java 1 log n 1 log n log n log n

Fibonacci heap IndexFibonacciMinPQ.java 1 log n ^† 1 1 ^† log n ^† 1

^† amortized guarantee

DATA STRUCTURE	CODE	INSERT	DEL-MIN	MIN	DEC-KEY	DELETE	MERGE
array	BruteIndexMinPQ.java	1	n	n	1	1	n
binary heap	IndexMinPQ.java	log n	log n	1	log n	log n	n
d-way heap	IndexMultiwayMinPQ.java	log_d n	d log_d n	1	log_d n	d log_d n	n
binomial heap	IndexBinomialMinPQ.java	1	log n	1	log n	log n	log n
Fibonacci heap	IndexFibonacciMinPQ.java	1	log n ^†	1	1 ^†	log n ^†	1
						^† amortized guarantee

Symbol tables.

The table below summarizes the order of growth of the running time of operations for a variety of symbol tables, as implemented in this textbook. It ignores leading constants and lower-order terms.

worst case average case

DATA STRUCTURE CODE SEARCH INSERT DELETE SEARCH INSERT DELETE

sequential search
(in an unordered list) SequentialSearchST.java n n n n n n

binary search
(in a sorted array) BinarySearchST.java log n n n log n n n

binary search tree
(unbalanced) BST.java n n n log n log n sqrt(n)

red-black BST
(left-leaning) RedBlackBST.java log n log n log n log n log n log n

AVL
AVLTreeST.java log n log n log n log n log n log n

hash table
(separate-chaining) SeparateChainingHashST.java n n n 1 ^† 1 ^† 1 ^†

hash table
(linear-probing) LinearProbingHashST.java n n n 1 ^† 1 ^† 1 ^†

^† uniform hashing assumption

		worst case	average case
DATA STRUCTURE	CODE	SEARCH	INSERT	DELETE	SEARCH	INSERT	DELETE
sequential search (in an unordered list)	SequentialSearchST.java	n	n	n	n	n	n
binary search (in a sorted array)	BinarySearchST.java	log n	n	n	log n	n	n
binary search tree (unbalanced)	BST.java	n	n	n	log n	log n	sqrt(n)
red-black BST (left-leaning)	RedBlackBST.java	log n	log n	log n	log n	log n	log n
AVL	AVLTreeST.java	log n	log n	log n	log n	log n	log n
hash table (separate-chaining)	SeparateChainingHashST.java	n	n	n	1 ^†	1 ^†	1 ^†
hash table (linear-probing)	LinearProbingHashST.java	n	n	n	1 ^†	1 ^†	1 ^†
					^† uniform hashing assumption

Graph processing.

The table below summarizes the order of growth of the worst-case running time and memory usage (beyond the memory for the graph itself) for a variety of graph-processing problems, as implemented in this textbook. It ignores leading constants and lower-order terms. All running times are worst-case running times.

PROBLEM ALGORITHM CODE TIME SPACE

path DFS DepthFirstPaths.java E + V V

shortest path (fewest edges) BFS BreadthFirstPaths.java E + V V

cycle DFS Cycle.java E + V V

directed path DFS DepthFirstDirectedPaths.java E + V V

shortest directed path (fewest edges) BFS BreadthFirstDirectedPaths.java E + V V

directed cycle DFS DirectedCycle.java E + V V

topological sort DFS Topological.java E + V V

bipartiteness / odd cycle DFS Bipartite.java E + V V

connected components DFS CC.java E + V V

strong components Kosaraju–Sharir KosarajuSharirSCC.java E + V V

strong components Tarjan TarjanSCC.java E + V V

strong components Gabow GabowSCC.java E + V V

Eulerian cycle DFS EulerianCycle.java E + V E + V

directed Eulerian cycle DFS DirectedEulerianCycle.java E + V V

transitive closure DFS TransitiveClosure.java V (E + V) V²

minimum spanning tree Kruskal KruskalMST.java E log E E + V

minimum spanning tree Prim PrimMST.java E log V V

minimum spanning tree Boruvka BoruvkaMST.java E log V V

shortest paths (nonnegative weights) Dijkstra DijkstraSP.java E log V V

shortest paths (no negative cycles) Bellman–Ford BellmanFordSP.java V (V + E) V

shortest paths (no cycles) topological sort AcyclicSP.java V + E V

all-pairs shortest paths Floyd–Warshall FloydWarshall.java V³ V²

maxflow–mincut Ford–Fulkerson FordFulkerson.java E V (E + V) V

bipartite matching Hopcroft–Karp HopcroftKarp.java V^½ (E + V) V

assignment problem successive shortest paths AssignmentProblem.java n³ log n n²

PROBLEM	ALGORITHM	CODE	TIME	SPACE
path	DFS	DepthFirstPaths.java	E + V	V
shortest path (fewest edges)	BFS	BreadthFirstPaths.java	E + V	V
cycle	DFS	Cycle.java	E + V	V
directed path	DFS	DepthFirstDirectedPaths.java	E + V	V
shortest directed path (fewest edges)	BFS	BreadthFirstDirectedPaths.java	E + V	V
directed cycle	DFS	DirectedCycle.java	E + V	V
topological sort	DFS	Topological.java	E + V	V
bipartiteness / odd cycle	DFS	Bipartite.java	E + V	V
connected components	DFS	CC.java	E + V	V
strong components	Kosaraju–Sharir	KosarajuSharirSCC.java	E + V	V
strong components	Tarjan	TarjanSCC.java	E + V	V
strong components	Gabow	GabowSCC.java	E + V	V
Eulerian cycle	DFS	EulerianCycle.java	E + V	E + V
directed Eulerian cycle	DFS	DirectedEulerianCycle.java	E + V	V
transitive closure	DFS	TransitiveClosure.java	V (E + V)	V²
minimum spanning tree	Kruskal	KruskalMST.java	E log E	E + V
minimum spanning tree	Prim	PrimMST.java	E log V	V
minimum spanning tree	Boruvka	BoruvkaMST.java	E log V	V
shortest paths (nonnegative weights)	Dijkstra	DijkstraSP.java	E log V	V
shortest paths (no negative cycles)	Bellman–Ford	BellmanFordSP.java	V (V + E)	V
shortest paths (no cycles)	topological sort	AcyclicSP.java	V + E	V
all-pairs shortest paths	Floyd–Warshall	FloydWarshall.java	V³	V²
maxflow–mincut	Ford–Fulkerson	FordFulkerson.java	E V (E + V)	V
bipartite matching	Hopcroft–Karp	HopcroftKarp.java	V^½ (E + V)	V
assignment problem	successive shortest paths	AssignmentProblem.java	n³ log n	n²

Commonly encountered functions.

Here are some functions that are commonly encountered when analyzing algorithms.

FUNCTION NOTATION DEFINITION

floor $ \lfloor x \rfloor $ greatest integer $\; \le \; x$

ceiling $ \lceil x \rceil $ smallest integer $\; \ge \; x$

binary logarithm $ \lg x$ or $\log_2 x$ $y$ such that $2^{\,y} = x$

natural logarithm $ \ln x$ or $\log_e x $ $y$ such that $e^{\,y} = x$

common logarithm $ \log_{10} x $ $y$ such that $10^{\,y} = x$

iterated binary logarithm $ \lg^* x $ $0$ if $x \le 1;\; 1 + \lg^*(\lg x)$ otherwise

harmonic number $ H_n $ $1 + 1/2 + 1/3 + \ldots + 1/n$

factorial $ n! $ $1 \times 2 \times 3 \times \ldots \times n$

binomial coefficient $ n \choose k $ $ \frac{n!}{k! \; (n-k)!}$

FUNCTION	NOTATION	DEFINITION
floor	\( \lfloor x \rfloor \)	greatest integer \(\; \le \; x\)
ceiling	\( \lceil x \rceil \)	smallest integer \(\; \ge \; x\)
binary logarithm	\( \lg x\) or \(\log_2 x\)	\(y\) such that \(2^{\,y} = x\)
natural logarithm	\( \ln x\) or \(\log_e x \)	\(y\) such that \(e^{\,y} = x\)
common logarithm	\( \log_{10} x \)	\(y\) such that \(10^{\,y} = x\)
iterated binary logarithm	\( \lg^* x \)	\(0\) if \(x \le 1;\; 1 + \lg^*(\lg x)\) otherwise
harmonic number	\( H_n \)	\(1 + 1/2 + 1/3 + \ldots + 1/n\)
factorial	\( n! \)	\(1 \times 2 \times 3 \times \ldots \times n\)
binomial coefficient	\( n \choose k \)	\( \frac{n!}{k! \; (n-k)!}\)

Useful formulas and approximations.

Here are some useful formulas for approximations that are widely used in the analysis of algorithms.

Harmonic sum: $1 + 1/2 + 1/3 + \ldots + 1/n \sim \ln n$
Triangular sum: $1 + 2 + 3 + \ldots + n = n \, (n+1) \, / \, 2 \sim n^2 \,/\, 2$
Sum of squares: $1^2 + 2^2 + 3^2 + \ldots + n^2 \sim n^3 \, / \, 3$
Geometric sum: If $r \neq 1$, then $1 + r + r^2 + r^3 + \ldots + r^n = (r^{n+1} - 1) \; /\; (r - 1)$
- $r = 1/2$: $1 + 1/2 + 1/4 + 1/8 + \ldots + 1/2^n \sim 2$
- $r = 2$: $1 + 2 + 4 + 8 + \ldots + n/2 + n = 2n - 1 \sim 2n$, when $n$ is a power of 2
Stirling's approximation: $\log_2 (n!) = \log_2 1 + \log_2 2 + \log_2 3 + \ldots + \log_2 n \sim n \log_2 n$
Exponential: $(1 + 1/n)^n \sim e; \;\;(1 - 1/n)^n \sim 1 / e$
Binomial coefficients: ${n \choose k} \sim n^k \, / \, k!$ when $k$ is a small constant
Approximate sum by integral: If $f(x)$ is a monotonically increasing function, then $ \displaystyle \int_0^n f(x) \; dx \; \le \; \sum_{i=1}^n \; f(i) \; \le \; \int_1^{n+1} f(x) \; dx$

Properties of logarithms.

Definition: $\log_b a = c$ means $b^c = a$. We refer to $b$ as the base of the logarithm.
Special cases: $\log_b b = 1,\; \log_b 1 = 0 $
Inverse of exponential: $b^{\log_b x} = x$
Product: $\log_b (x \times y) = \log_b x + \log_b y $
Division: $\log_b (x \div y) = \log_b x - \log_b y $
Finite product: $\log_b ( x_1 \times x_2 \times \ldots \times x_n) \; = \; \log_b x_1 + \log_b x_2 + \ldots + \log_b x_n$
Changing bases: $\log_b x = \log_c x \; / \; \log_c b $
Rearranging exponents: $x^{\log_b y} = y^{\log_b x}$
Exponentiation: $\log_b (x^y) = y \log_b x $

Asymptotic notations: definitions.

NAME NOTATION DESCRIPTION DEFINITION

Tilde $f(n) \sim g(n)\; $ $f(n)$ is equal to $g(n)$ asymptotically
(including constant factors) $ \; \displaystyle \lim_{n \to \infty} \frac{f(n)}{g(n)} = 1$
Big Oh $f(n)$ is $O(g(n))$ $f(n)$ is bounded above by $g(n)$ asymptotically
(ignoring constant factors) there exist constants $c > 0$ and $n_0 \ge 0$ such that $0 \le f(n) \le c \cdot g(n)$ for all $n \ge n_0$
Big Omega $f(n)$ is $\Omega(g(n))$ $f(n)$ is bounded below by $g(n)$ asymptotically
(ignoring constant factors) $ g(n) $ is $O(f(n))$
Big Theta $f(n)$ is $\Theta(g(n))$ $f(n)$ is bounded above and below by $g(n)$ asymptotically
(ignoring constant factors) $ f(n) $ is both $O(g(n))$ and $\Omega(g(n))$
Little oh $f(n)$ is $o(g(n))$ $f(n)$ is dominated by $g(n)$ asymptotically
(ignoring constant factors) $ \; \displaystyle \lim_{n \to \infty} \frac{f(n)}{g(n)} = 0$
Little omega $f(n)$ is $\omega(g(n))$ $f(n)$ dominates $g(n)$ asymptotically
(ignoring constant factors) $ g(n) $ is $o(f(n))$

NAME	NOTATION	DESCRIPTION	DEFINITION
Tilde	\(f(n) \sim g(n)\; \)	\(f(n)\) is equal to \(g(n)\) asymptotically (including constant factors)	\( \; \displaystyle \lim_{n \to \infty} \frac{f(n)}{g(n)} = 1\)
Big Oh	\(f(n)\) is \(O(g(n))\)	\(f(n)\) is bounded above by \(g(n)\) asymptotically (ignoring constant factors)	there exist constants \(c > 0\) and \(n_0 \ge 0\) such that \(0 \le f(n) \le c \cdot g(n)\) for all \(n \ge n_0\)
Big Omega	\(f(n)\) is \(\Omega(g(n))\)	\(f(n)\) is bounded below by \(g(n)\) asymptotically (ignoring constant factors)	\( g(n) \) is \(O(f(n))\)
Big Theta	\(f(n)\) is \(\Theta(g(n))\)	\(f(n)\) is bounded above and below by \(g(n)\) asymptotically (ignoring constant factors)	\( f(n) \) is both \(O(g(n))\) and \(\Omega(g(n))\)
Little oh	\(f(n)\) is \(o(g(n))\)	\(f(n)\) is dominated by \(g(n)\) asymptotically (ignoring constant factors)	\( \; \displaystyle \lim_{n \to \infty} \frac{f(n)}{g(n)} = 0\)
Little omega	\(f(n)\) is \(\omega(g(n))\)	\(f(n)\) dominates \(g(n)\) asymptotically (ignoring constant factors)	\( g(n) \) is \(o(f(n))\)

Common orders of growth.

NAME NOTATION EXAMPLE CODE FRAGMENT
Constant
$O(1)$
array access
arithmetic operation
function call
op();
Logarithmic
$O(\log n)$
binary search in a sorted array
insert in a binary heap
search in a red–black tree
for (int i = 1; i <= n; i = 2*i)
   op();
Linear
$O(n)$
sequential search
grade-school addition
BFPRT median finding
for (int i = 0; i < n; i++)
   op();
Linearithmic
$O(n \log n)$
mergesort
heapsort
fast Fourier transform
for (int i = 1; i <= n; i++)
   for (int j = i; j <= n; j = 2*j)
      op();
Quadratic
$O(n^2)$
enumerate all pairs
insertion sort
grade-school multiplication
for (int i = 0; i < n; i++)
   for (int j = i+1; j < n; j++)
      op();
Cubic
$O(n^3)$
enumerate all triples
Floyd–Warshall
grade-school matrix multiplication
for (int i = 0; i < n; i++)
   for (int j = i+1; j < n; j++)
      for (int k = j+1; k < n; k++)
         op();
Polynomial $O(n^c)$ ellipsoid algorithm for LP
AKS primality algorithm
Edmond's matching algorithm

Exponential $2^{O(n^c)}$ enumerating all subsets
enumerating all permutations
backtracking search

NAME	NOTATION	EXAMPLE	CODE FRAGMENT
Constant	\(O(1)\)	array access arithmetic operation function call	op();
Logarithmic	\(O(\log n)\)	binary search in a sorted array insert in a binary heap search in a red–black tree	for (int i = 1; i <= n; i = 2*i) op();
Linear	\(O(n)\)	sequential search grade-school addition BFPRT median finding	for (int i = 0; i < n; i++) op();
Linearithmic	\(O(n \log n)\)	mergesort heapsort fast Fourier transform	for (int i = 1; i <= n; i++) for (int j = i; j <= n; j = 2*j) op();
Quadratic	\(O(n^2)\)	enumerate all pairs insertion sort grade-school multiplication	for (int i = 0; i < n; i++) for (int j = i+1; j < n; j++) op();
Cubic	\(O(n^3)\)	enumerate all triples Floyd–Warshall grade-school matrix multiplication	for (int i = 0; i < n; i++) for (int j = i+1; j < n; j++) for (int k = j+1; k < n; k++) op();

Polynomial	\(O(n^c)\)	ellipsoid algorithm for LP AKS primality algorithm Edmond's matching algorithm
Exponential	\(2^{O(n^c)}\)	enumerating all subsets enumerating all permutations backtracking search

Asymptotic notations: properties.

Reflexivity: $f(n)$ is $O(f(n))$.
Constants: If $f(n)$ is $O(g(n))$ and $ c > 0 $, then $c \cdot f(n)$ is $O(g(n)))$.
Products: If $f_1(n)$ is $O(g_1(n))$ and $ f_2(n) $ is $O(g_2(n)))$, then $f_1(n) \cdot f_2(n)$ is $O(g_1(n) \cdot g_2(n)))$.
Sums: If $f_1(n)$ is $O(g_1(n))$ and $ f_2(n) $ is $O(g_2(n)))$, then $f_1(n) + f_2(n)$ is $O(\max \{ g_1(n) , g_2(n) \})$.
Transitivity: If $f(n)$ is $O(g(n))$ and $ g(n) $ is $O(h(n))$, then $ f(n) $ is $O(h(n))$.
Polynomials: Let $f(n) = a_0 + a_1 n + \ldots + a_d n^d$ with $a_d > 0$. Then, $ f(n) $ is $\Theta(n^d)$.
Logarithms and polynomials: $ \log_b n $ is $O(n^d)$ for every $ b > 0$ and every $ d > 0 $.
Exponentials and polynomials: $ n^d $ is $O(r^n)$ for every $ r > 0$ and every $ d > 0 $.
Factorials: $ n! $ is $ 2^{\Theta(n \log n)} $.
Limits: If $ \; \displaystyle \lim_{n \to \infty} \frac{f(n)}{g(n)} = c$ for some constant $ 0 < c < \infty$, then $f(n)$ is $\Theta(g(n))$.
Limits: If $ \; \displaystyle \lim_{n \to \infty} \frac{f(n)}{g(n)} = 0$, then $f(n)$ is $O(g(n))$ but not $\Theta(g(n))$.
Limits: If $ \; \displaystyle \lim_{n \to \infty} \frac{f(n)}{g(n)} = \infty$, then $f(n)$ is $\Omega(g(n))$ but not $O(g(n))$.

Here are some examples.

FUNCTION $o(n^2)$ $O(n^2)$ $\Theta(n^2)$ $\Omega(n^2)$ $\omega(n^2)$ $\sim 2 n^2$ $\sim 4 n^2$

$\log_2 n$ ✔ ✔

$10n + 45$ ✔ ✔

$2n^2 + 45n + 12$ ✔ ✔ ✔ ✔

$4n^2 - 2 \sqrt{n}$ ✔ ✔ ✔ ✔

$3n^3$ ✔ ✔

$2^n$ ✔ ✔

Divide-and-conquer recurrences.

For each of the following recurrences we assume $T(1) = 0$ and that $n\,/\,2$ means either $\lfloor n\,/\,2 \rfloor$ or $\lceil n\,/\,2 \rceil$.

RECURRENCE $T(n)$ EXAMPLE

$T(n) = T(n\,/\,2) + 1$ $\sim \log_2 n$ binary search

$T(n) = 2 T(n\,/\,2) + n$ $\sim n \log_2 n$ mergesort

$T(n) = T(n-1) + n$ $\sim \frac{1}{2} n^2$ insertion sort

$T(n) = 2 T(n\,/\,2) + 1$ $\sim n$ tree traversal

$T(n) = 2 T(n-1) + 1$ $\sim 2^n$ towers of Hanoi

$T(n) = 3 T(n\,/\,2) + \Theta(n)$ $\Theta(n^{\log_2 3}) = \Theta(n^{1.58...})$ Karatsuba multiplication

$T(n) = 7 T(n\,/\,2) + \Theta(n^2)$ $\Theta(n^{\log_2 7}) = \Theta(n^{2.81...})$ Strassen multiplication

$T(n) = 2 T(n\,/\,2) + \Theta(n \log n)$ $\Theta(n \log^2 n)$ closest pair

RECURRENCE	\(T(n)\)	EXAMPLE
\(T(n) = T(n\,/\,2) + 1\)	\(\sim \log_2 n\)	binary search
\(T(n) = 2 T(n\,/\,2) + n\)	\(\sim n \log_2 n\)	mergesort
\(T(n) = T(n-1) + n\)	\(\sim \frac{1}{2} n^2\)	insertion sort
\(T(n) = 2 T(n\,/\,2) + 1\)	\(\sim n\)	tree traversal
\(T(n) = 2 T(n-1) + 1\)	\(\sim 2^n\)	towers of Hanoi
\(T(n) = 3 T(n\,/\,2) + \Theta(n)\)	\(\Theta(n^{\log_2 3}) = \Theta(n^{1.58...})\)	Karatsuba multiplication
\(T(n) = 7 T(n\,/\,2) + \Theta(n^2)\)	\(\Theta(n^{\log_2 7}) = \Theta(n^{2.81...})\)	Strassen multiplication
\(T(n) = 2 T(n\,/\,2) + \Theta(n \log n)\)	\(\Theta(n \log^2 n)\)	closest pair

Master theorem.

Let $a \ge 1$, $b \ge 2$, and $c > 0$ and suppose that $T(n)$ is a function on the non-negative integers that satisfies the divide-and-conquer recurrence $$T(n) = a \; T(n\,/\,b) + \Theta(n^c)$$ with $T(0) = 0$ and $T(1) = \Theta(1)$, where $n\,/\,b$ means either $\lfloor n\,/\,b \rfloor$ or either $\lceil n\,/\,b \rceil$.

If $c < \log_b a$, then $T(n) = \Theta(n^{\log_{\,b} a})$
If $c = \log_b a$, then $T(n) = \Theta(n^c \log n)$
If $c > \log_b a$, then $T(n) = \Theta(n^c)$

Remark: there are many different versions of the master theorem. The Akra–Bazzi theorem is among the most powerful.

FUNCTION	\(o(n^2)\)	\(O(n^2)\)	\(\Theta(n^2)\)	\(\Omega(n^2)\)	\(\omega(n^2)\)	\(\sim 2 n^2\)	\(\sim 4 n^2\)
\(\log_2 n\)	✔	✔
\(10n + 45\)	✔	✔
\(2n^2 + 45n + 12\)		✔	✔	✔		✔
\(4n^2 - 2 \sqrt{n}\)		✔	✔	✔			✔
\(3n^3\)				✔	✔
\(2^n\)				✔	✔